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A191315 Sum of the heights of all dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0) steps at positive heights). 2
0, 0, 1, 2, 6, 11, 27, 50, 115, 216, 481, 913, 1992, 3809, 8192, 15748, 33512, 64685, 136546, 264422, 554686, 1077055, 2248105, 4375221, 9095238, 17735812, 36745504, 71776633, 148288346, 290092160, 597876033, 1171153370, 2408702852, 4723840544, 9697826974, 19038878297 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) = Sum_{k>=0} k * A191314(n,k).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.:  G(z) = Sum_{j>=0}(jz^(2j)/(F(j)F(j+1))), where F(k) are polynomials in z defined by F(0)=1, F(1)=1-z, F(k)=F(k-1)-z^2*F(k-2) for k>=2. The coefficients of these polynomials form the triangle A108299.

EXAMPLE

a(4)=6 because the sum of the heights of the paths HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD is 0+1+1+1+1+2=6; here U=(1,1), H=(1,0), D=(1,-1).

MAPLE

F[0] := 1: F[1] := 1-z: for k from 2 to 36 do F[k] := sort(expand(F[k-1]-z^2*F[k-2])) end do: G := sum(j*z^(2*j)/(F[j]*F[j+1]), j = 0 .. 34): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 35);

# second Maple program:

b:= proc(x, y, m) option remember;

      `if`(y>x or y<0, 0, `if`(x=0, m, b(x-1, y-1, m)+

      `if`(y=0, b(x-1, y, m), 0)+b(x-1, y+1, max(m, y+1))))

    end:

a:= n-> b(n, 0$2):

seq(a(n), n=0..30);  # Alois P. Heinz, Mar 13 2017

MATHEMATICA

b[x_, y_, m_] := b[x, y, m] = If[y > x || y < 0, 0, If[x == 0, m, b[x - 1, y - 1, m] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, May 16 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A108299, A191314.

Sequence in context: A007186 A033304 A091622 * A184884 A275222 A165821

Adjacent sequences:  A191312 A191313 A191314 * A191316 A191317 A191318

KEYWORD

nonn

AUTHOR

Emeric Deutsch, May 31 2011

STATUS

approved

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Last modified October 18 14:04 EDT 2019. Contains 328161 sequences. (Running on oeis4.)